Monday, 21 December 2020

Maxwell's equations have something missing

James C. Maxwell is peeved
I return to chapter one and Maxwell's equations.  Carroll's version of Maxwell's equations "in 19th century notation" at his equations 1.92 are$$
\nabla\times\mathbf{B}-\partial_t\mathbf{E}=\mathbf{J}$$
$$\nabla\bullet\mathbf{E}=\rho$$
$$\nabla\times\mathbf{E}+\partial_t\mathbf{B}=0 $$
$$\nabla\bullet\mathbf{B}=0$$
But that's not Maxwell's equations! In SI units they are

$$\nabla\times\mathbf{B}-\frac{1}{c^2}\partial_t\mathbf{E}=\mu_0\mathbf{J}$$
$$\nabla\bullet\mathbf{E}=\frac{\rho}{\epsilon_0}$$
$$\nabla\times\mathbf{E}+\partial_t\mathbf{B}=0$$
$$\nabla\bullet\mathbf{B}=0$$
Where ##\epsilon_0,\mu_0## are the electric and magnetic constants. Carroll has already announced that we were setting the speed of light ##c=1## but he does not say anything about ##\epsilon_0,\mu_0## here or later. They are just left out. I first noticed this when calculating the Energy-Momentum tensor for electro-magnetic radiation. I think that in Carrol's "natural units" (which might be a bit unorthodox) we also have ##\epsilon_0=1## and that means that, as ##c=1##, ##\mu_0=1## too.

See my reasoning in Commentary 1.8 Maxwells equations and units.pdf (two pages). 

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